Space-Efficient Fragments of Higher-Order Fixpoint Logic Florian Bruse 1 2 Martin Lange 1 Etienne Lozes 2 1 Universit¨ at Kassel, Germany 2 LSV, ENS Paris-Saclay, France September 8, 2017
Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic Motivation Higher-order Modal Fixpoint Logic: • extension of the modal µ -calculus by simply typed λ -calculus • very expressive logic, e.g., context-sensitive path languages, assume-guarantee properties • captures time hierarchy: type order k + 1 model checking is k + 1-EXPTIME complete (via alternating reachability game) 2 / 15
Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic Motivation Higher-order Modal Fixpoint Logic: • extension of the modal µ -calculus by simply typed λ -calculus • very expressive logic, e.g., context-sensitive path languages, assume-guarantee properties • captures time hierarchy: type order k + 1 model checking is k + 1-EXPTIME complete (via alternating reachability game) Question: Can we capture space hierarchy? 2 / 15
Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic Motivation Higher-order Modal Fixpoint Logic: • extension of the modal µ -calculus by simply typed λ -calculus • very expressive logic, e.g., context-sensitive path languages, assume-guarantee properties • captures time hierarchy: type order k + 1 model checking is k + 1-EXPTIME complete (via alternating reachability game) Question: Can we capture space hierarchy? → Tail-recursive fragment: • restrict interplay of fixpoints, mixing with certain operators • model-checking for order k + 1 in k -EXPSPACE (via nondet. reachability game) 2 / 15
Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic Motivation Higher-order Modal Fixpoint Logic: • extension of the modal µ -calculus by simply typed λ -calculus • very expressive logic, e.g., context-sensitive path languages, assume-guarantee properties • captures time hierarchy: type order k + 1 model checking is k + 1-EXPTIME complete (via alternating reachability game) Question: Can we capture space hierarchy? → Tail-recursive fragment: • restrict interplay of fixpoints, mixing with certain operators • model-checking for order k + 1 in k -EXPSPACE (via nondet. reachability game) • matching lower bound via reduction to order- k corridor tiling problem 2 / 15
Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic Motivation Higher-order Modal Fixpoint Logic: • extension of the modal µ -calculus by simply typed λ -calculus • very expressive logic, e.g., context-sensitive path languages, assume-guarantee properties • captures time hierarchy: type order k + 1 model checking is k + 1-EXPTIME complete (via alternating reachability game) Question: Can we capture space hierarchy? → Tail-recursive fragment: • restrict interplay of fixpoints, mixing with certain operators • model-checking for order k + 1 in k -EXPSPACE (via nondet. reachability game) • matching lower bound via reduction to order- k corridor tiling problem • still very expressive: a n b n c n path language 2 / 15
Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic Syntax of HFL HFL = modal µ -calculus ϕ ::= P | X | ϕ ∨ ϕ | ¬ ϕ | � a � ϕ | µ X .ϕ plus duals ϕ ∧ ψ , [ a ] ϕ , ν X natively 3 / 15
Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic Syntax of HFL HFL = modal µ -calculus + simply typed λ -calculus [Viswanathan 2 ’04] ϕ ::= P | X | x | ϕ ∨ ϕ | ¬ ϕ | � a � ϕ | µ X .ϕ | λ x .ϕ | ϕ ϕ plus duals ϕ ∧ ψ , [ a ] ϕ , ν X natively 3 / 15
Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic Syntax of HFL HFL = modal µ -calculus + simply typed λ -calculus [Viswanathan 2 ’04] ϕ ::= P | X | x | ϕ ∨ ϕ | ¬ ϕ | � a � ϕ | µ X .ϕ | λ x .ϕ | ϕ ϕ plus duals ϕ ∧ ψ , [ a ] ϕ , ν X natively But expressions like � a � P � b � P not well defined 3 / 15
Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic Syntax of HFL HFL = modal µ -calculus + simply typed λ -calculus [Viswanathan 2 ’04] ϕ ::= P | X | x | ϕ ∨ ϕ | ¬ ϕ | � a � ϕ | µ ( X : τ ) .ϕ | λ ( x : τ ) .ϕ | ϕ ϕ plus duals ϕ ∧ ψ , [ a ] ϕ , ν ( X : τ ) natively But expressions like � a � P � b � P not well defined well-formedness condition given by type system 3 / 15
Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic Types simple types given via τ ::= Pr complete lattice over transition system T = ( S , − → , L ) ] := (2 S , ⊆ ) [ [Pr] 4 / 15
Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic Types simple types given via τ ::= Pr | τ → τ because of right-associativity: τ = τ 1 → . . . → τ m → Pr each type induces a complete lattice over transition system T = ( S , − → , L ) using pointwise orderings ⊑ ] := (2 S , ⊆ ) [ [Pr] [ [ σ → τ ] ] := ([ [ σ ] ] → [ [ τ ] ] , ⊑ ) 4 / 15
Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic Semantics by Example � Consider ( µ X .λ x . x ∨ ( X [ a ] x ) P . 5 / 15
Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic Semantics by Example � Consider ( µ X .λ x . x ∨ ( X [ a ] x ) P . Unfolding via σ X .ψ = ψ [ σ X .ψ/ X ] yields � � µ X .λ x ′ . x ′ ∨ ( X [ a ] x ′ ) � � λ x . x ∨ [ a ] x P . 5 / 15
Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic Semantics by Example � Consider ( µ X .λ x . x ∨ ( X [ a ] x ) P . Unfolding via σ X .ψ = ψ [ σ X .ψ/ X ] yields � � µ X .λ x ′ . x ′ ∨ ( X [ a ] x ′ ) � � λ x . x ∨ [ a ] x P . Using β -reduction we get µ X .λ x ′ . x ′ ∨ ( X [ a ] x ′ ) � � P ∨ [ a ] P . 5 / 15
Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic Semantics by Example � Consider ( µ X .λ x . x ∨ ( X [ a ] x ) P . Unfolding via σ X .ψ = ψ [ σ X .ψ/ X ] yields � � µ X .λ x ′ . x ′ ∨ ( X [ a ] x ′ ) � � λ x . x ∨ [ a ] x P . Using β -reduction we get µ X .λ x ′ . x ′ ∨ ( X [ a ] x ′ ) � � P ∨ [ a ] P . More unfolding: P ∨ ( λ x ′ . x ′ ∨ µ X .λ x ′′ . x ′′ ∨ ( X [ a ] x ′′ ) [ a ] x ′ ) [ a ] P . � � 5 / 15
Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic Semantics by Example � Consider ( µ X .λ x . x ∨ ( X [ a ] x ) P . Unfolding via σ X .ψ = ψ [ σ X .ψ/ X ] yields � � µ X .λ x ′ . x ′ ∨ ( X [ a ] x ′ ) � � λ x . x ∨ [ a ] x P . Using β -reduction we get µ X .λ x ′ . x ′ ∨ ( X [ a ] x ′ ) � � P ∨ [ a ] P . More unfolding: P ∨ ( λ x ′ . x ′ ∨ µ X .λ x ′′ . x ′′ ∨ ( X [ a ] x ′′ ) [ a ] x ′ ) [ a ] P . � � More β -reduction: µ X .λ x ′′ . x ′′ ∨ ( X [ a ] x ′′ ) � � P ∨ [ a ] P ∨ [ a ][ a ] P ) . 5 / 15
Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic Semantics by Example � Consider ( µ X .λ x . x ∨ ( X [ a ] x ) P . Unfolding via σ X .ψ = ψ [ σ X .ψ/ X ] yields � � µ X .λ x ′ . x ′ ∨ ( X [ a ] x ′ ) � � λ x . x ∨ [ a ] x P . Using β -reduction we get µ X .λ x ′ . x ′ ∨ ( X [ a ] x ′ ) � � P ∨ [ a ] P . More unfolding: P ∨ ( λ x ′ . x ′ ∨ µ X .λ x ′′ . x ′′ ∨ ( X [ a ] x ′′ ) [ a ] x ′ ) [ a ] P . � � More β -reduction: µ X .λ x ′′ . x ′′ ∨ ( X [ a ] x ′′ ) � � P ∨ [ a ] P ∨ [ a ][ a ] P ) . We get: P ∨ [ a ] P ∨ [ a ][ a ] P ∨ . . . = � n i =0 [ a ] i P uniform inevitability! 5 / 15
Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic Tail-Recursion Limit use of fp vars: No free fixpoint variables in all subformulas of certain form • Form ϕ ∧ ψ : no free fixpoint variables in ϕ • Form [ a ] ϕ : no free fixpoint variables in ϕ • Form ¬ ϕ : no free fixpoint variables in ϕ • Form ψ ϕ : no free fixpoint variables in ϕ 6 / 15
Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic Tail-Recursion Limit use of fp vars: No free fixpoint variables in all subformulas of certain form • Form ϕ ∧ ψ : no free fixpoint variables in ϕ • Form [ a ] ϕ : no free fixpoint variables in ϕ • Form ¬ ϕ : no free fixpoint variables in ϕ • Form ψ ϕ : no free fixpoint variables in ϕ Result: free fixpoint variables not under any of these operators, each fixpoint definition “morally” contains only one variable (modulo nondeterminism), in operator position 6 / 15
Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic Tail-Recursion Limit use of fp vars: No free fixpoint variables in all subformulas of certain form • Form ϕ ∧ ψ : no free fixpoint variables in ϕ • Form [ a ] ϕ : no free fixpoint variables in ϕ • Form ¬ ϕ : no free fixpoint variables in ϕ • Form ψ ϕ : no free fixpoint variables in ϕ Result: free fixpoint variables not under any of these operators, each fixpoint definition “morally” contains only one variable (modulo nondeterminism), in operator position µ X . p ∨ ( � a � ( X ∧ q )) not tail-recursive 6 / 15
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